This was a good example that helped me to understand that you could use the optimization toolbox to determine Simulink model parameters from experimental data. However, I had many issues getting this script to work in v. 2008a. I would also recommend the reader to the demo script in the optimization toolbox, runtracklsq.m. This demo provides a simple example of using the optimization function lsqnonlin to determine Simulink model parameters to match a desired response. The documentation for this demo can be found in the online Matlab documentation under "Least Squares (Model Fitting) Examples."
I made the following changes to the script and model:
1. Removed the Stiffness multiplier on the force input. This multiplier violates the F=ma relationship of the second order model. The summing junction present in the model should be a summation of forces: input force, damping force, friction force, and spring stiffness force (due to position displacement).
2. The input and model constants present in the script cause the noise variance to swamp the response. The signal to noise ratio was also likely affected by the correction to the model noted in point (1). The noise variance is used to replicate experimentally derived data. This is a good component of the script for the user to learn how well the parameter identification is working. Increased U input force constant to 50 (force input) and reduced the spring constant, vksar, to 122. This increases the deflection of the second order system, relative to the given noise. Alternatively, the noise variance parameter, var, could be reduced.
3. Per the matlab example, runtracklsq, removed the subfunction LocalOptimize and changed the optimization function, LocalCostFcn, to a nested function. As it was, subfunctions do not share variable scope with their parent, so the mass and spring constants were not available to the simulation in the original optimization subfunction. Nested funtions are created by placing an end statement at the end of the nested function and at the end of the file to close out the parent function.
4. Removed the assignin('base') calls throughout the script as the nested function approach placed variables in a common 'current' workspace. The line that set the var noise variable back to zero was changed to simply:
var = 0.0000; % No artificaial noise in simulations
Added the following assignment operations in the LocalCostFcn to upload the optimization variable results to the model:
cv = varParamNum(1);
fd = varParamNum(2);
5. Removed the system input variable and made the 'model' system name a comment in the script for both the experimental and the truth model execution. Loaded the model at the beginning of the parent script, per the matlab runtracklsq.m example.
6. Changed the simulation output variables to the default [t, x, y]. Copied the first simulation run that modeled experimental data to a separate [yTest] variable so it could then be compared against the truth model in the optimation cost function. It seems that these are the default output variables and the copy operation is necessary to avoid data overwrites. As a result, the cost function F was changed to F = (y - yTest).
7. I also increased the friction constant to 10, just to add more nonlinearity to the model. However, this was not required to make the demo work.
8. Added the following plot command to the optimization function to overlay experimental vs. model time histories: plot(t,yTest, 'b', t, y,'r');
I can see the generic elegance of the author's original approach with the assignin operators and vectors for fixed and variable parameters. However, within the time I had available to me, I could not get this approach to work.
